Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/169

Rh therefore the double of the third is greater than the double of the fourth, and therefore the third is greater than the fourth.

In the same manner, if the first be equal to the second, or less than it, the third may be shewn to be equal to the fourth, or less than it.

Wherefore, if the first &c.

PROPOSITION B. THEOREM.

If four magnitudes he proportionals, they shall also be proportionals when taken inversely.

Let A be to B as C is to D: then also, inversely, B shall be to A as D is to C.

Take of B and D any equimultiples whatever E and F; and of A and G any equimultiples whatever G and H. First, let E be greater than G, then G is less than E. Then, because A is to B as C is to D; [Hypothesis. and of A and C the first and third, G and H are equimultiples; and of B and D the second and fourth, E and F are equimultiples; and that G is less than E; therefore H is less than F; [V. Def. 5. that is, F is greater than H. Therefore, if E be greater than G, F is greater than H.

In the same manner, if E be equal to G, F may be shewn to be equal to H and if less, less.

But E and F are any equimultiples whatever of B and D, and G and H are any equimultiples wliatever of A and C; [Construction therefore B is to A as D is to C. [V. Definition 5.

Wherefore, if four magnitudes &c.